Practical Work #4

EFREI – M1 IFM Numerical analysis applied to financial issues

Practical Work #4

The Finite Difference Method (FDM) has been presented in the course. We aim at applying this method to the well-known Black & Scholes equation with constant volatilityσand constant interest rater for the modelling of a European vanilla put option:







∂P˜

∂t (t,S)+σ²S² 2

∂²P˜

∂S²(t,S)+r S∂P˜

∂S(t,S)−rP˜(t,S)=0, P˜(T,S)=max(0,K−S),

or equivalently (by means of the change of variablesP(t,S)=P˜(T −t,S))

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



∂P

∂t(t,S)−σ²S² 2

∂²P

∂S²(t,S)−r S∂P

∂S(t,S)+r P(t,S)=0, (1a)

P(0,S)=max(0,K−S). (1b)

We set for the present study

K=100,T =1,σ=0.2 and r=0.04 . IfPis a solution to (1), then the following function

ϕ(θ,x)=P(θ,e^x)e^{−αθ−βx} with β=1 2− r

σ²,α= −r−σ²β² 2 , is a solution to







∂ϕ

∂θ−σ² 2

∂²ϕ

∂x² =0, (2a)

ϕ(0,x)=e^−βx·max(0,K−e^x). (2b)

We recall that the exact solution is given by

P˜(t,S)=K e^−r(T−t)Φ(−d2)−SΦ(−d1), withΦ(d)= 1

p2π Z d

−∞

exp µ

−z² 2

¶

dzand:

d₁=ln(S/K)+(r+σ²/2)(T −t) σp

T −t ,d₂=d₁−σp T −t.

We thus aim at simulating equivalent formulations (1) and (2) and then at comparing corresponding solu- tions.

08/04/2013 1 M. Girardin – B. Meltz

EFREI – M1 IFM Numerical analysis applied to financial issues

Exercise 1 (Heat equation on a uniform grid) Let x and x be two real numbers such that x¿lnK ¿x.

We consider a space discretization given by xj =x+(j−1)∆x,∆x= x−x

N_x−1 for some Nx >1and a time discretization tn=(n−1)∆t ,∆t= T

Nt−1for a suitable Nt>1.

Implement and compare performance of the explicit Euler scheme, the implicit Euler scheme and the Crank- Nicholson scheme for the resolution of (2). Comparisons will be made on the primitive functionP .˜

Note that the stability condition reads∆t≤∆x² σ² .

Exercise 2 (Heat equation on a nonuniform grid) Let S and S be two real numbers such that S¿K ¿S.

We consider an asset discretization given by Sj =S+(j−1)∆S,∆S= S−S

NS−1 for some NS >1and a time discretization tn=(n−1)∆t ,∆t= T

Nt−1for some Nt>1. The corresponding space discretization is imposed by the change of variable:

xj=lnSj. We set∆xj=xj+1−xj.

1. Derive a formula approaching the second order spatial derivative on nonuniform grids.

2. Then implement the resolution of (2)by means of the Crank-Nicholson scheme.

Exercise 3 (Resolution of the Black & Scholes model in primitive variables)

1. Given a uniform discretization of the time-asset space [0,T]×[S,S], propose a numerical scheme based on the explicit Euler scheme for the time derivative to solve(1).

2. Find out by means of numerical simulations a suitable value for∆t . 3. Implement the Crank-Nicholson scheme applied to(1).

4. Comment numerical results.

08/04/2013 2 M. Girardin – B. Meltz